Geotechnical Engineering

DHARM

366 GEOTECHNICAL ENGINEERING

Since the principal stresses are known from equations 10.30 and 10.31, the maximum

shear stress τmax may be obtained as:

τmax =

()

.sin

σσ

π

(^13) θ 0
2
−
=F
HG
I
KJ
q
...(Eq. 10.34)
This will attain its highest value when θ 0 = 90°, which equals q/π.
∴τabsolute maximum =
q
π
...(Eq. 10.35)
This value, it is easily understood, occurs at points lying on a semi-circle of diameter
equal to the width of the strip, B. Hence the maximum shear stress under the centre of a
continuous strip occurs at a depth of B/2 beneath the centre.
The knowledge of shear stresses may not be important in normal foundation design
procedure, but Jürgenson (1934) obtained the solution for this case. Pressure bulbs of shear
stress as obtained by him are shown in Fig. 10.11.
B=2b q/unit area
0.50 q/p
0.75 q/p
0.95 q/p
q/p
0.90 q/p
0.80 q/p
0.70 q/p
0.60 q/p
0.50 q/p
0.40 q/p
0.1 q/p
0.5 B
0.30 q/p
1.0 B
1.5 B
2.0 B
Fig. 10.11 Pressure bulbs of shear stress under
strip load (after Jürgenson, 1934)

10.5 Uniform Load on Circular Area

This problem may arise in connection with settlement studies of structures on circular founda-
tions, such as gasoline tanks, grain elevators, and storage bins.
The Boussinesq equation for the vertical stress due to a point load can be extended to
find the vertical stress at any point beneath the centre of a uniformly loaded circular area. Let
the circular area of radius a be loaded uniformly with q per unit area as shown in Fig. 10.12.
Let us consider an elementary ring of radius r and thickness dr of the loaded area. This
ring may be imagined to be further divided into elemental areas, each δA; the load from such
an elemental area is q. δA. The vertical stress δ σz at point A, at a depth z below the centre of
the loaded area, is given by: